Problem: Solve for $x$ and $y$ using elimination. ${-3x-6y = -42}$ ${4x+5y = 41}$
Solution: We can eliminate $y$ by adding the equations together when the $y$ coefficients have opposite signs. Multiply the top equation by $5$ and the bottom equation by $6$ ${-15x-30y = -210}$ $24x+30y = 246$ Add the top and bottom equations together. $9x = 36$ $\dfrac{9x}{{9}} = \dfrac{36}{{9}}$ ${x = 4}$ Now that you know ${x = 4}$ , plug it back into $\thinspace {-3x-6y = -42}\thinspace$ to find $y$ ${-3}{(4)}{ - 6y = -42}$ $-12-6y = -42$ $-12{+12} - 6y = -42{+12}$ $-6y = -30$ $\dfrac{-6y}{{-6}} = \dfrac{-30}{{-6}}$ ${y = 5}$ You can also plug ${x = 4}$ into $\thinspace {4x+5y = 41}\thinspace$ and get the same answer for $y$ : ${4}{(4)}{ + 5y = 41}$ ${y = 5}$